[FOM] Throwing Darts, Time, and the Infinite

Thomas Forster T.Forster at dpmms.cam.ac.uk
Wed Apr 18 07:37:07 EDT 2012

This is an old chestnut.  I am taking the liberty of inflicting
on listmembers the following email from my colleague Imre leader, with his 
permission. (slightly edited).

Dear Thomas,

I was thinking about that thing you told me, that was supposedly
against CH: that one bijects $\Re$ with the set of countable ordinals
and then throws one dart and then another at the board.

My reply then was (correctly) that this was just silly, as it confuses
what conditional probability means (one cannot condition on an event of
zero probability, as in the phrase `given that I throw $\alpha$'), and
also it forgets that not all sets are measurable.

That reply was entirely correct, but I have had two further thoughts. The
first one is: the fact that the event `second dart beats first' is in
fact one of the absolutely most standard examples of a non-measurable set.
Indeed (assuming CH) one takes the subset of $[0,1] \times [0,1]$ given by
those points $\tuple{x,y}$ for which $x<y$ in the ordering induced from
$\omega_1$. Then each row is countable but each column is cocountable!

My second thought is more important. It is that, even ignoring the
fact that the `paradox' is rubbish because of conditional probability
and nonmeasurable sets, even then, it is {\bf not} against CH. What I
mean is, I will hereby run the {\bf exact} same paradox without any CH
assumption. Ready? Here goes \ldots

Let $\kappa$ be the least cardinality of a set of positive measure. Let
such a set be $A$, and let us well-order $A$ in such a way that all
initial segments are smaller than kappa. [he means $\kappa$-like]. Note
that all initial segments have measure zero, by definition of $\kappa$.

OK, our experiment is: throw a dart at the set $A$. Twice. [Note: as $A$
has positive measure, we can do this by throwing a dart at $[0,1]$ and 
only counting it when it lands in $A$.] Then all the paradox still 

Conclusion: there is no way, not even intuitively or anything, that this
`paradox' has anything to do with CH.


On Tue, 17 Apr 2012, Jeremy Gwiazda wrote:

> Hello,
> Chris Freiling?s Axioms of Symmetry have, I believe, been discussed on
> FOM at least twice. In ?Axioms of Symmetry: Throwing Darts at the Real
> Number Line?, Freiling considers two darts thrown at [0, 1]. He
> writes, ?the real number line does not really know which dart was
> thrown first or second?, which leads to one of his axioms of symmetry.
> In a recently published paper, I suggest that a well-ordering of [0,
> 1] does know the order of the darts under certain assumptions. Fix a
> well-ordering of [0, 1]. Let r1 be the real hit by the first dart.
> Then assuming ZFC and CH, there are only countably many reals less
> than r1 in the well-ordering. Thus with probability 1 the second dart
> hits a real greater than r1 in the well-ordering. (Put again slightly
> differently: working in ZFC, Freiling demonstrates that assuming that
> the reals can?t tell the order of the darts proves not CH; I argue
> that assuming CH means that a well-ordering of [0, 1] can tell the
> order of the darts.) I go on to create a puzzle using special
> relativity. In case it is of interest, the paper is here:
> http://www.springerlink.com/content/e2746w1kn6913580/
> An earlier, countable version of a similar puzzle is available here:
> http://www.springerlink.com/content/v1n12t200jv2553u/
> Best,
> Jeremy Gwiazda
> jgwiazda at gc.cuny.edu
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom


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